The puzzling magnetoresistance of graphite

At the Journal Club on Condensed Matter, Jason Alicea has a stimulating commentary on an interesting experimental paper,

Two Phase Transitions Induced by a Magnetic Field in Graphite
Benoît Fauqué, David LeBoeuf, Baptiste Vignolle, Marc Nardone, Cyril Proust, and Kamran Behnia 

In the experiment, the temperature dependence of the resistance of graphite [stacks of layers of graphene] is measured with magnetic fields perpendicular to the layers for fields up to 80 Tesla.
[n.b. the high field! This is a pulsed field.]
It is estimated that above 7.5 Tesla all the graphene layers are in their lowest Landau level.

The figure below shows a colour shaded plot of the interlayer resistance [on a logarithmic scale] as a function of temperature and magnetic field.

This is highly suggestive of a phase diagram and that several phase transitions occur as a function of magnetic field. This is highly appealing (seductive?) because there are theoretical predictions [reviewed by Alicea] of such phase transitions, specifically into charge density wave states (CDW).
Furthermore, there are even proposals of (sexy) edge states in directions perpendicular to the layers.

However, I am not so sure. I remain to be convinced that there are thermodynamic phase transitions. The experiment is observing changes in interlayer transport properties.
Here are my concerns.

1. The conduction is metallic within the layers and insulating perpendicular to the layers. This is clearly seen in the figure below showing resistivity versus temperature at a field of 47 Tesla. In particular, note that the resistance parallel to the layers [red squares and left linear scale] is essentially temperature independent. In contrast, the resistivity perpendicular to the layers [blue squares and right logarithmic scale] becomes activated, below about 6 K, with an energy gap of about
2.4 meV=25 K.

As Alicea points out, this is very strange. Normally, a charge gap (like in a quasi-one-dimensional CDW) will produce activated transport in all directions.

2. Even at lower fields, i.e. below 10 Tesla, the interlayer magnetoresistance of graphite is anomalous. In a simple Fermi liquid, there is no orbital magnetoresistance when the magnetic field and current are parallel. Pal and Maslov consider this problem of longitudinal magnetoresistance. With a simple model calculation using a realistic band structure for graphite they obtain a saturating magnetoresistance of about 20 percent, whereas the experimental value is orders of magnitude larger. 

3. A large interlayer magnetoresistance when the magnetic field is parallel to the current (and so there is no Lorentz force) is reminiscent of unusual magnetoresistance seen in a wide range of correlated electron materials.

4. Confusion about "insulating" magnetoresistance has occurred before. I discussed this earlier in the context of the large orbital magnetoresistance of a transition metal oxide PdCoO2. Just because the resistance increases with decreasing temperature at a fixed field does not mean that the system is in an insulating state. Copper does this! From my point of view, this was one of the problems with the exotic claims made by Anderson, Clarke, and Strong in the 1990s of non-Fermi liquid states in Bechgaard salts [see for example this PRL and Science paper]. They never actually calculated magnetoresistance but made claims about field-induced "confinement" phase transitions. Understanding these angle-dependent magnetoresistance experiments remains an open problem.

5. The results are sample dependent, although qualitatively consistent. Different results are obtained for different forms of graphite (kish and HOPG). Furthermore, in layered metals, it is difficult to measure the interlayer and intralayer resistivity. Due to impurities, stacking defects, and contact misalignment the electrical current path will inevitably contain some mixture of interlayer and intra-layer paths.

6. I am a bit confused about the argument concerning edge states. The authors suggest that the fact that the interlayer resistivity saturates at low temperatures (i.e. is no longer activated) is evidence for edge states. It is not clear to me why there is no evidence of the gap in the interlayer resistance. They say something about the mixing of interlayer and intralayer currents due to stacking defects, but I did not find it a particularly clear argument.

So what is the most likely explanation? I am not sure. Perhaps, there are no phase transitions. The role of the field may be to somehow decouple the layers. There are several comparable energy scales involved, particularly due to the semi-metal character of graphite.

Thermodynamic experiments are desirable. Calculations of the interlayer and interlayer resistance within CDW models need to be done too.